## Abstract

Segmented mirror telescope designs address issues of mechanical rigidity, but introduce the problem of aligning, or cophasing, the separate segments to conform to the optimum mirror shape. While several solutions have been widely adopted, a few difficulties persist — the introduction of non-common path errors and an artificial division of the problem into coarse and fine regimes involving separate dedicated hardware solutions. Here we propose a novel method that addresses many of these issues. Fizeau Interferometric Cophasing of Segmented Mirrors (FICSM) uses non-redundant sparse aperture interferometry to phase mirror segments to interferometric precision using unexceptional science hardware. To show the potential of this technique we numerically simulate conditions on NASA’s James Webb Space Telescope (JWST), showing that the FICSM method has the potential to phase the primary mirror from an initial state with segment-to-segment pistons as large as 150 microns and tilts as large as 0.5 arcseconds, to produce a final state with 0.75 nm rms segment-to-segment pistons and 3.7 mas rms segment tilts. The image undergoes monotonic improvement during this process. This results in a rms wavefront error of 3.65 nm, well below the 100 nm requirement of JWST’s coarse phasing algorithm.

© 2012 OSA

## 1. Introduction

The drive to telescopes with ever larger apertures has led to an explosion of interest in segmented mirror designs which can significantly reduce problems with mechanical rigidity. However, mirror segmentation introduces a new problem: that of aligning the separate segments to closely match the optimum mirror shape. This cophasing process is critical for the delivery of successful observational science. Many of the world’s largest telescopes and all of the next generation of Extremely Large Telescopes plan to employ a segmented design, making the development and improvement of cophasing methods vital.

We posit that each segment has 6 rigid-body degrees of freedom (3 translational positions and 3 rotations), but of these, 3 are considered stabilized by the segment mounting system and generally do not require alignment to optical precision, or are aligned through other means. The remaining degrees of freedom are the position perpendicular to the plane of the mirror (piston) and two rotations around axes in the plane of the mirror (tip/tilt). The foundation of any segmented optical telescope is the detection and correction of these aberrations. The problem of segment rotation about the normal to the center of the segment (called clocking) and the degeneracy between clocking and the azimuthal position of the segment are not treated here. These problems are amenable to other solutions [1].

It is important to make clear from the outset that segment cophasing falls under the category of active optics, which involves slow or infrequent corrections to match the mirror to the optimum shape. This contrasts with adaptive optics, which seeks to minimize the fast effects associated with the turbulent atmosphere. Segment cophasing is often tasked with bringing the segments into alignment from large offsets, rather than continuously measuring and maintaining their position. The Keck telescopes, the most well known telescopes with segmented primary mirrors, are phased only after a segment has been swapped or when the segment alignments are expected to have degraded substantially; phasing is not part of routine observing.

Some cophasing techniques are used routinely on telescopes. Among the most common are modified Shack-Hartmann sensing [2, 3] and modified curvature sensing [4, 5], and both have shown great success. These approaches are generalizations of traditional wavefront sensing techniques, utilizing physical optics to model the effects of diffraction from a discontinuous surface. However, these methods suffer from some disadvantages and are far from ideal. Both utilize dedicated phasing cameras, so the primary mirror is optimized for a camera that is not used for science. Some require relative motion between the primary mirror and the camera, increasing the complexity of the operation.

Our proposed method is Fizeau Interferometric Cophasing of Segmented Mirrors (FICSM), first developed in Cheetham 2011 [6]. We utilize the increasingly widespread technique of Sparse Aperture Interferometry [7, 8] to develop a conceptually and operationally simple cophasing plan that avoids many of the disadvantages mentioned above. In particular, Non-Redundant Aperture Masking (NRM) is used here to illustrate the method, but the process is general and applicable to any form of sparse aperture interferometry.

It should be stressed that NRM allows cophasing only for those mirrors behind the mask holes. Options to use NRM to cophase an entire mirror include rotating the mask or utilizing multiple aperture masks. Another option to implement FICSM involves tilting primary mirror segments to different pointing origins, generating interference patterns from entire segments, known as segment tilting interferometry. In this paper, we deal solely with the application of FICSM to NRM imaging, building the basics of the method around one of the simplest applications of sparse aperture interferometry.

By fragmenting the telescope pupil into a non-redundant array of subapertures, every set of interference fringes can be traced to a unique pair of subapertures, corresponding to unique regions on the primary mirror. Information obtained from the fringes present in the images can then be used to infer the state of the mirror segments to interferometric precision.

An *N* hole aperture mask exhibits *n _{b}* =

*N*(

*N*− 1)/2 baselines (or hole pairs). For the case of a non-redundant array geometry, each baseline produces a fringe pattern at a unique spatial frequency, hence the number of baselines is greater than the number of holes when

*N*> 3. The distinguishable fringe patterns comprising such an interferogram are most readily studied in the Fourier plane (or image spectrum), where the amplitude and phase at any given point for which there is power can be termed the complex visibility on that specific baseline. The square of the absolute value of the image spectrum is referred to as the power spectrum, and we note that NRM often does not fill the Fourier plane completely, in which case only a subset of the available spatial frequencies passed by the unocculted primary mirror are measured by a single NRM image.

In order to illustrate the potential of FICSM as a cophasing strategy, we explore an application to the JWST. As a segmented mirror space telescope, there is the added risk of failure of dedicated hardware. Both currently adopted cophasing techniques rely on the NIRCam instrument, so a failure of that instrument would be catastrophic to the mission. Alternative cophasing methods must use existing hardware on JWST, and be able to phase the mirror from an initial state with a maximum of 100 *μ*m segment pistons to a state with a final root-mean-square (rms) wavefront error of 100 nm to maximize telescope performance [9]. To comfortably test this range, we opt for a maximum initial segment-to-segment piston error of 150 *μ*m.

A non-redundant aperture mask is planned for the NIRISS instrument [10], allowing this science camera to be used with FICSM to cophase JWST. However, combining FICSM with segment tilting interferometry would allow all mirror segments to be phased using any of the imaging science cameras planned for JWST [11].

#### 1.1. Phasing segmented mirrors with NRM

A non-redundant mask has the special property that each baseline vector is unique, and therefore produces power in a unique and separately isolated region of the Fourier plane. This has two critical consequences. Firstly information extracted from interferograms is unambiguous: A Fourier datum can be uniquely identified with the baseline from which it arose. This is not the case for a redundant aperture, where this mapping is an ill-conditioned inverse problem, and therefore possesses no general solution. Following Lloyd 2006 [12], we refer to the helpfully segregated areas of signal in the Fourier domain delivered by NRM as *splodges*. Secondly, the information extracted from each splodge is uniquely related to a pair of mask holes, and therefore to two regions on the primary mirror.

These features make it immediately apparent that an NRM-inspired approach offers a novel solution to the problem of phasing segmented primary mirror telescopes. Information extracted from interferograms taken with the science camera can be directly related to the phase structure of specific points on the wavefront at the pupil. By placing the mask such that each hole exposes the surface of a single primary mirror segment, one exposure can deliver independent information about the phases of all exposed segments simultaneously. Moreover, since the number of hole pairs (or baselines) usually greatly exceeds the number of holes, a single image provides multiple measurements to constrain the cophasing of each mirror. Due to the over-constrained nature of this mapping between baseline phases and the mirror glass, errors due to wild data points can be identified and eliminated by enforcing self-consistency on the solution.

At the outset of our cophasing problem all segments are assumed to have unknown errors in both tip/tilt and piston. It is important that our cophasing strategy is able to identify and separate the effects of these two errors, because under some circumstances they can produce similar signals. The key to separating tip/tilt from piston-induced errors in our method is to use images taken with both a narrow and a broad optical bandwidth filter. The two different cophasing error terms then exhibit quite distinct behaviors and can be readily discriminated, as described below.

Our method utilizes three observables readily extracted from the splodges in the interferogram image spectrum: (1) the baseline phase; (2) the fringe power; and (3) the phase ramp or slope across the splodge. The effects of piston and tilt on these three observables extracted from an NRM image are illustrated in Fig. 1. Without any aberrations, interferograms give high contrast fringes (as seen in 1a, 1g) with zero phase (1d, 1j). After applying piston errors, the fringes move and (for the broadband case) begin to blur (1b,1h), resulting in a constant phase (in the narrowband case, 1e) or radial phase slopes (in the broadband case, 1k) across the corresponding Fourier splodges. After applying tilts to the mirrors, the centroids of the Airy patterns from each aperture move, yet the fringes remain centered where the patterns overlap (as seen in 1c, 1i). This causes a linear phase ramp across the corresponding splodge in the Fourier plane regardless of bandwidth (1f,1l). Unlike the case for piston however, these slopes are not necessarily radial, instead following the direction of the arbitrary tilt errors.

Table 1 summarizes the expected effect on the observables recovered from analysis of any given splodge in the image spectrum for a given type of alignment error of the mirror. Perhaps the most critical thing to note is that for the case of a monochromatic image, the effects of tilt and piston are entirely independent and separable. A monochromatic image, or a narrowband image (e.g. using a filter with 1% fractional bandwidth) will allow tilt to be measured in isolation directly from the phase ramps across each splodge in the Fourier plane. After removing tilts, a broadband image (e.g. using a filter with 4.5% fractional bandwidth) will allow measurement of the pistons from the shape of the amplitude and phase distributions in the Fourier plane. As we show below, this step has a remarkably wide single step capture range. Finally, the fringe phase measured at the center of the splodge can be used for fine alignment to sub-wavelength precision.

## 2. Mathematical basis

In order to understand the processes behind FICSM, it is important to have an analytical model of the optical setup. The intensity distribution at the detector can be derived using simple Fourier optics. A diffraction limited imaging system is assumed, with the only optical aberrations considered being piston and tip/tilt applied to sub-apertures. Additionally, the imaging camera is assumed to well sample the interference fringes at the wavelengths used.

Consider the telescope pupil as being partitioned into N identical non-overlapping subapertures with uniform and identical reflectivity. We observe a point source through a filter such that the spectrum of the incident light is described by *f*(*λ*) (energy per wavelength interval per square meter). The combined intensity distribution on the detector expressed in terms of the field (*E*) at each wavelength will be given by Eq. (1) below. We define (*ξ*, *η*) to be the coordinates in the image plane (expressed as angles on the sky), and (*x*, *y*) to be the coordinates in the pupil plane (in meters).

*λ*

_{1}and

*λ*

_{2}are the extremal wavelengths of the filter’s bandpass.

By the principle of superposition, the field *E* can be expressed as a sum of the fields *E _{i}* from each subaperture. We write the shape of each subaperture as

*C*(

_{i}*x*−

*x*,

_{i}*y*−

*y*), where the hole center is (

_{i}*x*,

_{i}*y*), and write its piston as

_{i}*p*and tip/tilt (expressed as mirror gradients) as (

_{i}*m*,

_{i}*n*) in the (

_{i}*x*,

*y*) directions. For circular holes,

*C*(

_{i}*x*−

*x*,

_{i}*y*−

*y*) would be a uniform disk with radius

_{i}*r*. We can then write each

_{i}*E*as the Fourier transform (ℱ) of the subaperture function, which becomes:

_{i}In order to evaluate this Fourier Transform we can use the Shift Theorem for Fourier Transforms [13]. We use it twice; once to incorporate the shifting due to the location of the subaperture away from the origin, and once in reverse to include the linear phase ramp term inside the Fourier transform caused by tip/tilt.

This leads to the analytical result for *E _{i}* shown in Eq. (3).

*A*(

_{i}*ξ*−

*m*,

_{i}*η*−

*n*) is the Fourier Transform of the subaperture function

_{i}*C*, shifted in the image plane (relative to the centroid of the image intensity distribution) due to the tilt of mirror

_{i}*i*. The phase term due to piston is unaffected by the Fourier Transform.

In the special case of circular holes, we can use the familiar result for the Fourier transform of a uniform disk shown in Eq. (4). When using other subaperture shapes this term must be replaced by the appropriate Fourier transform. Hexagonal segments, or shapes with an arbitrary number of straight edges, possess analytical Fourier transforms [14]. The phase terms will remain the same.

In general, the intensity distribution can then be written as the sum of the patterns generated from each pair of subapertures (with ^{*} denoting a complex conjugate):

*E*are given by Eq. (3).

_{i}The *I _{i,j}* terms then correspond to the fringes generated from interference between subapertures

*i*and

*j*. In the special case

*i*=

*j*this reduces to a form of the familiar Airy pattern shifted due to the tilt on the mirror associated with that subaperture. In the case

*i*≠

*j*, this gives interference fringes etched into the overlap of the two shifted Airy patterns from subapertures

*i*and

*j*, with an added phase due to the difference in piston.

We can express the interference terms in the image plane intensity, *I _{i,j}*, using En. 6:

*i*=

*j*, giving the familiar Airy pattern.

Writing the intensity distribution in this form immediately shows the effect of piston and tip/tilt on the interference fringes. A piston difference between the two mirrors will shift their fringe off the image’s intensity centroid. This fringe phase shift scales with wavelength. For a nonzero filter bandwidth, fringes at different wavelengths will no longer add in phase since they will be shifted away from the origin by different distances. This results in a decrease in fringe amplitude and a fringe blurring known as bandwidth smearing, as seen in Fig. 1. Since the difference between the two extremal wavelengths *λ*_{1} and *λ*_{2} is small compared to the wavelengths themselves, the difference in phase between the fringes changes over a much larger scale, characterized by the coherence length of the filter,

*λ*is the full width at half maximum (FWHM) of

*f*(

*λ*) and

*λ*= (

_{c}*λ*

_{1}+

*λ*

_{2})/2. Since the fringes are affected by relative piston only, the zero point of the pistons is unspecified. It can be chosen to be convenient for the descriptions of segment positioning.

## 3. The basis of the FICSM algorithm

In principle, a single interferogram taken through a non-redundant mask with a filter of nonzero bandwidth is sufficient to measure both tilt and piston aberrations. However these two aberrations can deliver similar signatures, and in the presence of other real world imperfections such as readout noise and telescope jitter, a robust strategy to extract cophasing information requires further complexity. The modifications required will depend upon the specific application (e.g. levels of noise, specifics of filters available etc), and must be developed through simulation. In the following sections we illustrate the FICSM method by applying it to JWST. Its use in other contexts should require further (straightforward) modifications.

Both mathematical simulation and physical intuition tell us that in the monochromatic case, two of our three observables – fringe power and phase slopes – are sensitive only to tilt errors (as seen in Table 1). These may therefore be measured in isolation and corrected to high precision, so that polychromatic images may then be employed to determine piston errors alone. Because of the complex way in which the shape of both the subapertures and the filter bandpass modify the observables, it turns out that in general a unique signal is imprinted on the phase and amplitude structure which may be recorded out to very large piston offsets. Capture ranges of tens to hundreds of microns (several coherence lengths) are enabled by the complex, non-repeating nature of these signals. At the final step, measuring the phase itself (rather than phase slopes) empowers sensitive metrology to achieve fine-adjustment of segments which are already cophased to within one wavelength.

#### 3.1. Measuring tilts

Tilts may be measured from a single monochromatic (in practice narrowband) image in two steps: a coarse measurement for each baseline, and a fine measurement for each subaperture. A coarse step is performed to ensure the fitting procedure in the fine step converges to the global minimum. The phase slope for any splodge is governed by the average of the tilts of the two mirrors that comprise that baseline. This is due simply to the fact that the interference fringes are located halfway between the two subaperture diffraction patterns, and position in the image plane is directly related to phase slope in the Fourier plane. This makes measuring the tilts relatively simple: the phase slope across each splodge is measured, immediately yielding a baseline mean tilt. The ensemble of measurements for each baseline are then converted to tilt estimates for each subaperture. The mathematical procedure to accomplish this is also used in identical context for piston measurements, and we have therefore dedicated a separate Section 3.3 to discuss it.

In order to refine the measurements, a second fine-tuning step is performed using least squares fitting. The fitting is performed on the images themselves rather than the image spectrum to reduce computational time, since Eqs. 5 and 6 give a direct way to compute the intensity distribution of the image. Non-linear least squares fitting is then performed using the goodness of fit parameter ( ${\chi}_{I}^{2}$) shown in Eq. (8). The unknown parameters for this fit are the mirror tilts and the constant phase offsets for each baseline caused by piston. Each iteration requires generating a new simulated image. This fitting is straightforward; the solution is a clear global minima, and generally requires approximately 10 iterations. This fine-tuning step also relies indirectly on the phase slope.

The tilt measurement process relies on the interference fringes, and so the capture range of this step is the maximum separation of the subaperture diffraction patterns that still allows some measurable interference between them.

#### 3.2. Measuring piston

This procedure is more involved than that for measuring the tilts, due to the complicated relationship between the complex visibilities and piston. A two step process is again employed, consisting of a coarse measurement for each baseline and a fine measurement for each subaperture, measured relative to the chosen reference (in this case, one of the segments). This is motivated by the two different scales resulting from the effects of piston; changing the piston on the scale of a wavelength changes the phase across each splodge by an approximately uniform amount and leaves the amplitude roughly the same, while changing the piston on the scale of a coherence length changes the shape of both the phase and amplitude function across each splodge. The role of the filter coherence length here is similar to that used in understanding the limits of positioning tolerances in the case of free-flying interferometers [15].

The coarse measurement procedure consists of comparing the phase and amplitude distributions in the image spectrum to a “lookup table” of distributions covering a range of pistons. Calculations are done in the Fourier plane, so that measurements for each baseline can be done separately. The data in the lookup table is generated once and saved prior to the phasing process, since it will depend only on the specifics of the instrumentation (mask and camera optics as well as the bandpass of the filter). This lookup table contains the complex visibility of each point on each splodge as a function of piston. The entries must be spaced by less than one wavelength to ensure that the fine step converges to the correct solution. In practice, a sampling density of 4 simulations per wavelength has been found to be sufficient.

The actual phase at any spatial frequency changes much too quickly as a function of piston for a coarse, large capture range measurement. To overcome this problem, only relative phase is used. The phase at the center of each splodge is subtracted from the entire splodge. This process is reminiscent of differential phase measurements common in interferometry, however in this case the signal is entangled with both spatial structure and phase slopes due to residual tilts. For each baseline, this modified version of the visibility distribution (*V*) of the splodge is compared to each theoretically generated one from the lookup table using the goodness-of-fit parameter (
${\chi}_{V}^{2}$) shown in Eq. (9). The (*u*, *v*) coordinates refer to the frequencies in the (*ξ*, *η*) directions of the image.

In order to ensure that the coarse estimate is accurate, this process is performed independently for two images in different observing bands. The visibility distributions at each piston will depend on the filter bandwidth and central wavelength, and so adding the ${\chi}_{V}^{2}$ distributions of the two images effectively eliminates near-degeneracies which can be present for a single filter alone, resulting in a much more robust coarse piston estimate. The point of minimum total ${\chi}_{V}^{2}$ for each baseline is then recorded as the coarse measured piston difference.

Since the equations for the image depend only on the piston differences, the “zero” piston position is arbitrary and undefined. For this reason, the piston for one of the mirrors is set to zero, and the other mirrors measured relative to it. In practice, any condition on the zero piston position could be set, including setting the average piston to be zero to minimize mirror motion. The piston differences measured for each baseline then need to be converted to measurements for each hole. This procedure is similar to that for tilts, and is described in section 3.3

After the coarse phasing step described above, a nonlinear least-squares fine fitting procedure is applied to one of the interferograms, with the piston values as the only unknown parameters. The coarse value is used as a starting parameter for the fit, and the procedure and misfit statistic for fitting is the same as for tilts. This procedure generally converges in approximately 10 iterations, and requires generating new simulated images each iteration.

This fine fitting step is equivalent to finding the absolute phase of each splodge in the Fourier plane, or lining up the fringe peaks in the image plane. This results in the *χ*^{2} space having local minima centered on the correct piston and spaced by *λ*. In order for this step to converge to the correct solution (so that the correct fringes line up), the coarse measurement must be accurate to within half a wavelength. Occasionally this condition is not met, leading to a final piston measurement that is an integral number of wavelengths (usually 1) from the actual value. While this is a rare occurrence (observed in one out of 100 complete simulations below), they are easily recognized and rectified.

In order to unambiguously identify these errors, the two images used for the coarse piston measurement can both be used in the fine step. The fine piston measurement will always ensure that the piston is measured to within an integral number of wavelengths. Since the two images are at different wavelengths, a disagreement between the measured values indicates that one or both are inaccurate. Only when both measurements are correct will they agree. Knowledge of the two wavelengths also allows us to immediately recover the actual piston in the case where they do disagree. The difference between the piston measurements will be equal to *cλ*_{2} − *dλ*_{1}, where *c* and *d* are unknown integers (usually −1, 0 or 1). This equation has a unique solution when assuming small *c* and *d*, and these can then be multiplied by the wavelength and subtracted from the measured piston to reconstruct the actual piston.

#### 3.3. Turning baseline data into estimates for mirrors

The phasing strategy described above requires a method for converting quantities recovered from each baseline into estimates for each subaperture. As previously discussed, the number of baselines is generally greater than the number of subapertures so that one image provides multiple constraints for each mirror. Enforcing consistency on both the tip/tilt and piston measurements dramatically reduces the impact of outliers and measurement errors. In order to turn the baseline measurements into estimates for each subaperture, a method to calculate the least squares solution is required. We have employed the method used in the Keck telescope cophasing algorithm [2], which is based on a singular value decomposition from Press et al. [16].

We have a system of equations relating the pistons on each mirror *p _{i}* to the measured piston difference

*δ*between mirrors

_{i,j}*j*and

*i*:

Similarly, turning to the tip/tilt problem, we have equations for the measured tip/tilt *M _{i,j}* for each baseline in terms of that on each mirror

*t*:

_{i}For the remainder of the method, pistons and tilts are treated identically, and so only the equations for pistons are shown. The equations for tilts are given by replacing *δ* with *M* and replacing *p* with *t*.

For an N hole mask, these are systems of
${n}_{b}=\frac{N\left(N-1\right)}{2}$ equations in *N* unknowns, and can be expressed in matrix form as:

Singular value decomposition allows the matrix *A* to be expressed in the form *A* = *Uwv ^{T}*, where

*w*is a diagonal matrix with diagonal elements (

*w*). This leads to the least squares solutions for piston, given by [16]:

_{j}*w*) is the diagonal matrix with diagonal elements 1/

_{j}*w*.

_{j}As mentioned above, the fact that there are more measurements than unknowns provides a way of checking the consistency of the measurements. Systems to accomplish this have been implemented for both piston and tilts. They work by combining measurements from several baselines to get an estimate of what the measurement for a particular baseline should be. All distinct combinations are calculated, and the actual measurement is replaced by the median measurement if it is beyond two wavelengths from the rest.

For example, an estimate for the measured piston difference *δ _{i,j}* can be calculated from two other baselines, since:

## 4. Cophasing JWST: a numerical case-study

Quantitative metrics of the success of this technique (such as capture range and residual error) require a specific case with real world error sources and limitations to be modeled, otherwise it is mathematically possible to cophase in a single step with arbitrary precision. A simulator was written in the IDL programming language capable of modeling the JWST primary mirror and producing images subject to various imperfections and noise processes. Our cophasing algorithm then performed the following steps:

- Using the specifications of the telescope, optics and detector, the coarse piston lookup table is computed and loaded.
- The initial state of the mirror is prepared by applying a random tip/tilt and piston to each segment.
- Cophasing begins by taking a narrow bandwidth image using the current mirror state.
- The image is processed with the tip/tilt fitting program, and the best-estimate tilts corrected in the present pupil.
- Images in two different broad bandwidth filters are taken.
- The two broad band images are processed with the piston fitting program, and the results compared.
- If they agree, the present pupil is corrected for the mean of the two measurements.
- If they disagree, the two measurements are used to reconstruct the true piston, which is then used to correct the pupil.
- Steps 3 – 8 are repeated once.
- The final fit residual pistons are computed by comparison with Step 2.

Here the “narrow” bandwidth image would ideally approach monochromatic, but realizing that we must work with commonly-available optics, we have developed the strategy so that a 1% fractional bandwidth can easily be made to work in simulations. Although 1% may sometimes be too broad to be approximated as monochromatic for the purpose of tilt fitting (increasing misfit residuals), a relatively straightforward cure is to repeat the fitting sequence as indicated at Step 9. Simulations have shown this to deliver very robust results, incurring a proportional penalty in increased observing time and algorithmic complexity. Turning to the broadband filters, fractional bandwidths as small as 4.5% have been shown to work in simulations. These numbers are rough guides, but the width of the narrow bandwidth filter will affect the accuracy of the tip/tilt fitting, while the widths of the broad bandwidth filters will define the capture range and accuracy of the piston fitting by changing the coherence length. The final accuracy of the piston fitting will also depend on the central wavelengths of the broad bandwidth filters.

To summarize, the entire cophasing process defined above requires one narrow and two broad band images with a tilt and a piston move, all performed twice — a total of 6 images and 4 segment moves, assuming perfect segment actuation. With real hardware an iterative process might be required to assert desired mirror moves with sufficient accuracy.

#### 4.1. Numerical simulations: setup and configuration

In order to explore the capability for the FICSM algorithm to cophase the JWST primary mirror, extensive numerical simulations have been performed. The method was implemented in the IDL programming language. Non-linear least squares fitting steps were achieved using MPFIT, a freely available IDL routine that uses the Levenberg-Marquardt technique. We have chosen the current specifications of JWST’s NIRCAM instrument with filter profiles taken from the WebbPSF software (Perrin et al. 2012 [17]). To generate the narrowband image, the F405N filter profile was used, while the broadband images used the F430M and F480M filter profiles. The mask used in the simulations is based on one already designed for JWST and described in Sivaramakrishnan et. al. [18], with a few small changes (0.6 m diameter circular holes were used instead of 0.8 m diameter hexagonal holes). The discrepancy in setup for our simulations was motivated only by clarity for the images and plots produced: similar outcomes should apply to both cases. The mask used for the simulations is shown in Fig. 2, along with a theoretical image generated using it, while basic data for our simulation setup is given in Table 2.

As stated in section 1, the requirements for a JWST phasing system are a capture range of more than 100 *μ*m in piston, and a final wavefront error of less than 100 nm. To comfortably test the viability of this method, a maximum piston capture range of 150 *μ*m was adopted, measured between any two segments. Using the coherence length of 96 *μ*m from the first broadband filter, this corresponds to more than 3 coherence lengths at the wavefront (6 coherence lengths for the second filter). We adopted a reasonable error budget of a maximum 0.5 arcseconds of tilt, chosen through consideration of the plate scale, 65 mas per pixel. This corresponds to more than 7.5 pixels, and is an estimate of the residual tilts from initial alignment steps rather than an exploration of the capture range of FICSM.

A number of important noise processes were also incorporated into the simulations, namely flat field errors, image jitter and photon noise. The level of noise for each of these processes was set using baseline design specifications for the JWST instrument [9]. The noise levels were further adjusted over a range around the nominal performance levels so as to explore the expected cophasing accuracy under various scenarios. It is our belief that we have captured the dominant contributions to degrading the accuracy of our algorithm in a flight context. Our specific choice of NIRCAM for these simulations is illustrative; as discussed briefly in section 5 this technique could be used with the NIRCam short wavelength, NIRCam long wavelength, and MIRI imaging cameras aboard JWST. NIRISS’ narrowest bandpass at wavelengths where the image is Nyquist-sampled is of the order of 5%. Further study of the three-filter approach described here will be required to determine the practicability of using NIRISS to perform coarse phasing.

#### 4.2. Numerical simulations: results

Both piston and tilt fitting were tested with a fixed capture range by using a random uniform distribution to generate the error terms. This also ensured that the maximum capture range for piston (set by the size of the lookup table) was not exceeded.

The results from running 100 complete phasing simulations under nominal noise conditions can be seen in Fig. 3. They used 0.2 pixels (13 mas) of jitter, a standard deviation of 0.1% for flat field errors, and photon noise from 10^{9} photons. Errors due to inaccuracy in segment actuator motion were not considered. The cophasing algorithm produced a final rms residual piston of 0.75 nm, and an rms residual tilt of 3.7 mas, showing that we can expect the method to work to well within the accuracy required from the JWST coarse phasing system. By comparison with the 65 mas plate scale of the detector, a tilt of this size would result in a misalignment on the detector of less than 6% of a pixel.

The corresponding total rms wavefront error is 3.65 nm, corresponding to a Strehl ratio of 0.99997 at 4.3 *μ*m using the common approximation *S* = *e*^{−(2πσ/λ)2}, where *σ* is the rms wavefront error and *λ* is the operating wavelength. This is more than an order of magnitude lower than the 100 nm requirement for coarse phasing the JWST primary mirror, and appears to be achievable with existing hardware and most of its science cameras.

Out of 100 simulations, only one simulation was flagged as having possibly inaccurate results after comparing the piston measurements at the two wavelengths. Both filters were a single wavelength off for each measurement in that simulation, but since the wavelengths were known the correct pistons were reconstructed. However, this proved to be substantially less accurate than directly measuring the piston, so these points skewed the results. Despite this, the maximum residual piston was less than 6 nm and the wavefront error from that simulation was 11 nm, significantly less than the 100 nm requirement. However, these measurements were detected by the algorithm. A strategy involving a further application of the method whenever this occurs would reduce the residual piston errors to levels similar to the remainder of the simulations, thereby reducing the rms residual piston to less than 0.1 nm.

It is also clear that the tilt residuals do not limit the accuracy of the piston measurements. Plotting the piston residuals against the tilt residuals yields no significant correlation.

## 5. Extending FICSM to other configurations

While our simulations have demonstrated the utility of FICSM with NRM, the technique is applicable to any form of sparse aperture interferometry. Since the ultimate goal is to phase segmented mirrors efficiently without the need for dedicated hardware, removing the need for an aperture mask would be ideal. Segment tilting interferometry is a maskless extension of NRM which promises the potential to phase entire mirror arrays. Segment tilting interferometry has demonstrated success in mid-infrared interferometric imaging at the Keck I telescope [8]. It works by deliberately repointing subsets of mirror segments to focus at different points within the field of view of the camera. By coaligning and cophasing selected groups of segments to the same pointing origin, sparse-aperture non-redundant interferograms can be generated without the need for a physical mask. The FICSM technique could then be applied to each of these interference patterns, thereby cophasing the subset of segments selected by the pattern. Repeating this process with differing mirror combinations, taking care to ensure that the segment patterns have some segments in common (so as to stitch the absolute phases to a common surface) would permit the cophasing of any segmented mirror. A more complete discussion is contained in [6]. Details of this stitching procedure are telescope- and hardware-dependent. For JWST the operational procedures already developed for stacking all single-segment PSFs at one location in the focal plane [9] can be used, with little modification, to stitch the distinct segment patterns together.

As presented here, FICSM has been specifically developed for telescopes such as JWST where it is possible to retrieve Fourier phase information directly from the interferograms. However, this process is made much more difficult when observing on large ground based telescopes through the turbulent atmospheric phase screen, particularly for the longest baselines. Similarly, unexpected large image jitter during JWST’s early commissioning stages might require tailoring the subsets of segments that form interferograms to have short baselines, so that the FICSM algorithm can still be used to cophase the telescope. Despite this, options for extending FICSM to work on seeing limited telescopes may be possible; indeed the earlier Keck segment tilting interferometry has shown it can work in the mid-infrared. A phasing strategy which can be employed on ground based segmented telescopes may prove particularly useful for the next generation of Extremely Large Telescopes (ELTs), but determining the efficacy of FICSM applied to the many hundreds of segments in some ELT designs is beyond the scope of this paper.

## 6. Conclusions

We present a new technique christened Fizeau Interferometric Cophasing of Segmented Mirrors (FICSM) for cophasing segmented mirror telescopes. The method uses a sparse aperture interferometric approach to measure the piston and tilt aberrations of mirror segments to interferometric precision. Feasibility has been demonstrated with numerical modeling of mirror cophasing applicable to the James Webb Space Telescope with the NIRCAM instrument. Our simulations started with the mirror in an initial state with large errors in piston and tilt (150 *μ*m and 0.5 arcsecond respectively) and incorporated realistic noise sources, yet delivered final residuals of a few nanometers in piston and 10 milliarcseconds of segment tilt – an improvement of more than 5 orders of magnitude. The method can be carried out using any of JWST’s scientific imaging cameras. Results were achieved after two iterations through a procedure requiring three exposures and two mirror adjustments each pass. Operational details of the technique follow existing procedures. We conclude that the FICSM technique has the potential to cophase the JWST primary mirror to more than an order of magnitude better than its coarse phasing requirements, and may also be useful to future segmented-mirror telescopes.

## Acknowledgments

This work was supported in part by the U.S. National Science Foundation grant AST-0804417, NASA grant APRA08-0117, and the STScI Director’s Discretionary Research Fund. We also wish to thank the anonymous reviewers for their helpful suggestions to improve the paper.

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